US6646593B1  Process for mapping multiplebounce ghosting artifacts from radar imaging data  Google Patents
Process for mapping multiplebounce ghosting artifacts from radar imaging data Download PDFInfo
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 US6646593B1 US6646593B1 US10059416 US5941602A US6646593B1 US 6646593 B1 US6646593 B1 US 6646593B1 US 10059416 US10059416 US 10059416 US 5941602 A US5941602 A US 5941602A US 6646593 B1 US6646593 B1 US 6646593B1
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 G—PHYSICS
 G01—MEASURING; TESTING
 G01S—RADIO DIRECTIONFINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCEDETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
 G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
 G01S13/88—Radar or analogous systems specially adapted for specific applications
 G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
 G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. correcting range migration errors
 G01S13/9035—Particular SAR processing techniques not provided for elsewhere, e.g. squint mode, doppler beamsharpening mode, spotlight mode, bistatic SAR, inverse SAR

 G—PHYSICS
 G01—MEASURING; TESTING
 G01S—RADIO DIRECTIONFINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCEDETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
 G01S15/00—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
 G01S15/88—Sonar systems specially adapted for specific applications
 G01S15/89—Sonar systems specially adapted for specific applications for mapping or imaging
 G01S15/8906—Shortrange imaging systems; Acoustic microscope systems using pulseecho techniques

 G—PHYSICS
 G01—MEASURING; TESTING
 G01S—RADIO DIRECTIONFINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCEDETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
 G01S15/00—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
 G01S15/88—Sonar systems specially adapted for specific applications
 G01S15/89—Sonar systems specially adapted for specific applications for mapping or imaging
 G01S15/8906—Shortrange imaging systems; Acoustic microscope systems using pulseecho techniques
 G01S15/8977—Shortrange imaging systems; Acoustic microscope systems using pulseecho techniques using special techniques for image reconstruction, e.g. FFT, geometrical transformations, spatial deconvolution, time deconvolution

 G—PHYSICS
 G01—MEASURING; TESTING
 G01S—RADIO DIRECTIONFINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCEDETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
 G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
 G01S13/88—Radar or analogous systems specially adapted for specific applications
 G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
 G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. correcting range migration errors
 G01S13/9035—Particular SAR processing techniques not provided for elsewhere, e.g. squint mode, doppler beamsharpening mode, spotlight mode, bistatic SAR, inverse SAR
 G01S2013/9052—Spotlight mode

 G—PHYSICS
 G01—MEASURING; TESTING
 G01S—RADIO DIRECTIONFINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCEDETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
 G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
 G01S7/52—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00
 G01S7/52017—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00 particularly adapted to shortrange imaging
 G01S7/52077—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00 particularly adapted to shortrange imaging with means for elimination of unwanted signals, e.g. noise or interference
Abstract
Description
This application claims priority to and incorporates by reference in its entirety, U.S. Provisional Patent Application No. 60/345,639, entitled “SPOTLIGHT SAR IMAGE FORMATION WITHOUT MULTIPLEBOUNCE GHOSTING ARTIFACTS” filed Jan. 8, 2002.
1. Field of the Invention
Generally, the invention involves a process for forming radar images. More specifically, the invention involves a process for mapping multiple bounce ghosts (MBGs) and for forming radar images without the deleterious effects of MBGs.
2. Description of the Related Art
Radar, at its most basic application, is used to measure the range to a target. With knowledge of the speed of propagation of the wave, i.e., electromagnetic wave, that is transmitted toward the target, it is possible to resolve in a first dimension, the distance to the target, based on the received reflected wave or echo. In order to use radar as an imaging tool, it is necessary to collect information about the crossrange of the target, in addition to the first dimension information. This crossrange information is about a second dimension perpendicular to the first dimension.
Synthetic aperture radar (SAR) can be used to collect data in both the first and second dimensions, through a process wherein the reflected waves are measured at different angles with respect to an objectofinterest. This process is referred to in the art as collecting radar measurements over a synthetic (as opposed to a literal) aperture. By taking various measurements of the objectofinterest from varying aspect angles, it is possible to determine approximate distance to the scattering centers within an objectofinterest in the first dimension and location of these scattering centers within the objectofinterest in the second, crossrange dimension. This process of twodimensional imaging is commonly referred to as reflection tomography.
SAR systems take advantage of the longrange propagation characteristics of radar signals and the complex information processing capability of modern digital electronics to provide highresolution imagery. SAR imaging is not restricted by time of day or atmospheric conditions due to its operative frequencies. Consequently, SAR imaging supplements other photographic and optical imaging techniques in order to facilitate environmental monitoring, earthresource mapping, and military operations which may require broadarea imaging at high resolutions. More specifically, SAR technology provides detailed terrain information to geologists for mineral exploration, environmentalists for determination of oil spill boundaries, navigators for sea state and ice hazard mapping, and the military for reconnaissance and targeting information.
Other systems using reflection data, also referred to as projection measurements, are fault inspection systems using acoustic imaging, submarine sonar for imaging underwater objects and the like, seismic imaging system for tunnel detection, oil exploration, geological surveys, etc., and medical diagnostic tools such as sonograms and echocardiograms.
There have been two basic types of processing techniques used in the field of reflection tomography to reconstruct singlebounce (SB) reflection data. First, the frequencydomain projectionslice theorem takes the measured phase history from the reflection data taken at different aspect angles and generates the reconstruction of an image using Fourier transforms. This reconstruction technique is often used for reconstructing SAR image data in order to minimize the computational load that results from necessarily complex processing. A second technique, more prevalent in the medical imaging community, is based on the timedomain back projection techniques. Both of these techniques are discussed in U.S. Pat. No. 5,805,098 to McCorkle which is incorporated herein by reference in its entirety.
The reflection data processing techniques described in the related art assume that the impinging wave reflects a single time off of the object of interest before returning back to the receiver. This assumption neglects the situation wherein the wave actually reflects off of multiple portions of the object of interest, cascading for any number of times, before returning to the receiver. All prior art on reflection tomography assumes only one bounce. That is, conventional tomography does not include the effects of multiplebounce (MB) scattering events wherein the mediating waveform first scatters off of one portion of the extended objectofinterest, which then scatters in a cascade fashion off of one or more other regions of this same extended objectof interest or off of other objects before scattering into the receiver. These facts motivate the need for the development of a modified image formation process that applies for cases in which the measured reflection data also identifies and accounts for MB scattering events.
A particular imaging scenario wherein the prior art process is insufficient includes determining the physical distribution of the scattering centers for cases in which multiplebounce echoes occur. For example, prior art processes would have a difficult time determining from three (3) received echoes that instead of three scattering centers for an objectofinterest, the image actually contained two targets, wherein the third echo was between the two scattering centers as opposed to a third scattering center. The problem is exacerbated in a case where there are many scattering centers within the image, with the possibility for multiple bounces between the scattering centers. Further, in the case of SAR imaging, the process becomes even more complicated by the fact that the object (or region)ofinterest radar information is only gathered over a very small aspect angle range, for example, up to approximately 15 degrees. Current processing schemes place the multiplebounce echoes at incorrect (ghost) locations due to fundamental assumptions implicit in the processing.
Described herein are frequencydomain backprojection processes for forming, e.g spotlight SAR images that are not corrupted by the effects of multiplebounce ghosting artifacts. These processes give an approximately exact reconstruction of the multiple bounce reflectivity function (MBRF) ƒ(x,y,γ). As shown in the embodiments herein, the reconstruction process is not affected to any greater extent by noise and interference than in the case of the single bounce processing. Specifically, the evaluation of ƒ(x,y,γ) in the γ=0 plane gives an approximately exact reconstruction of the true object scattering centers which is uncorrupted by multiplebounce contributions to the phase history data G(ξ,θ). In addition, the nonzero dependence of ƒ(x,y,γ) upon the MB coordinate γ can be used to facilitate the identification of featuresinterest within the imaged region.
In an embodiment of the present invention, a process is described for incorporating the effects of multiple bounces obtained using the back projection imaging process referred to herein, in order to obtain a cleaned up image that has removed the misleading effects of the multiple bounces, commonly referred to as multiple bounce ghosts (MBGs). The process identifies the MBGs and maps the MBG reflections into a metricallycorrect image plane, also referred to as, a delay image plane or higher dimensional image plane, creating auxiliary images, which are useful in providing further information about the viewing region, including additional discriminating information that could assist in object recognition and identification. The imaging information, including both single and multiple bounce information, is only required for a narrow range of measurement angles.
The processes of the present invention are useful in three dimensional imaging, providing images that are free of multipath effects, and extending the fieldofuse of the technology beyond SAR imaging to all areas of reflection tomography. Because the technology incorporates a physical model that explicitly allows for multibounce effects in image formation, it is applicable to any image formation technology where that phenomenon is found such as, but not limited to, real aperture radar imaging, synthetic aperture radar (SAR) imaging, inverse SAR (ISAR) imaging, active sonar underwater acoustic mapping, active geographic acoustic exploration and ultrasonic medical imaging.
In the Figures:
FIGS. 1(a)1(b) illustrate scattering events according to embodiments of the present invention;
FIGS. 2(a)2(d) illustrate true scattering point data from multiple depth planes, according to a first embodiment of the present invention;
FIGS. 3(a)3(b) illustrate measurement functions for scattering data according to a first embodiment of the present invention;
FIG. 4 illustrates scattering point data for an objectofinterest utilizing single bounce reconstruction, according to a first embodiment of the present invention;
FIGS. 5(a)5(d) illustrate scattering point data for an objectofinterest utilizing multiple bounce reconstruction according to a first embodiment of the present invention;
FIGS. 6(a)6(d) illustrate true scattering point data from multiple depth planes, according to a second embodiment of the present invention;
FIGS. 7(a)7(b) illustrate measurement functions for scattering data according to a second embodiment of the present invention;
FIG. 8 illustrates scattering point data for an objectofinterest utilizing single bounce reconstruction, according to a second embodiment of the present invention; and
FIGS. 9(a)9(d) illustrate scattering point data for an objectofinterest utilizing multiple bounce reconstruction according to a second embodiment of the present invention.
In a preferred embodiment of the present invention, the imaging process assumes the condition that a scattered waveform has the same characteristics as the incident waveform. In the context of SAR imaging, the measured range to target (i.e., assumed objectofinterest) of a given multiple bounce (MB) scattering event at some aperture angle is identical to that of an equivalent single bounce (SB) scattering event. FIG. 1a presents an example of such equivalent scattering events. In this example, the transmitter is colocated with the receiver (both shown as 10) at a large distance from the regionofinterest, so that the effects of a curved wave front can be neglected. One skilled in the art recognizes that the process described herein may be adjusted for non colocated transmitter and receiver configurations, as well as for distances from the region of interest which result in curved wavefronts. FIG. 1a shows that waveform traverses a total distance of 2L in propagating from the transmitter at 10 to scattering point (SP) 1 and back to the colocated receiver at 10. This distance represents the measured rangeto target from the colocated transmitterreceiver to SP 1 based upon waveform transmission and the corresponding reception of the its echo.
The specifics of the transmitter and receiver configurations usable with the embodiments described herein are known to those skilled in the art. Further, one skilled in the art recognizes that the mode of collection of the SAR data is not limited to any single mode, but may include, among others, spotlight mode, strip map mode, range Doppler profiler (RDP) mode, airtoair mode, and the like.
Still referring to FIG. 1a, the multiple scattering event due to SPs 2 and 3 are considered. Here, the transmitted waveform reflects off of SP 2, which then reflects off of SP 3 prior to reception. If the sum of the line segments A+B+C is equal to 2L in FIG. 1a, then the measured rangeto target of the singlescattering event at SP 1 at the shown measurement angle θ is identical to, and therefore indistinguishable from, the doublescattering event due to the cascaded reflection from SP 2 to SP 3. In addition, multiplescattering events involving three or more scattering points can have this same rangetotarget, as shown in FIG. 1b.
Referring to FIG. 1b, the sum of the twopoint scattering event, A+B+C, cascaded from SP 2 and SP 3 and measured at the receiver 10, is equivalent to the sum of the threepoint scattering event, D+E+F+G, cascaded from SP 4, SP 5, and SP 6 measured at the receiver 10, at the shown measurement angle θ.
As described in the Summary of the Problem, conventional SAR theory assumes that all scattering events which affect the measured phase history data (i.e., rangetotarget projections) are due to SB effects, as occurs with SP 1 in FIG. 1a. This is not always the case. When MB events, such as SPs 2 and 3 in FIG. 1a, also significantly impact the measured phase history data, the application of the conventional SAR image formation techniques yield imperfect reconstructions of the regionofinterest.
According to an embodiment of the present invention, the best image of the true scattering centers is obtained by taking into account the MBG data retrieved via imaging through application of an image reconstruction algorithm for multibounce scattering (IRAMS). This process for applying the IRAMS is developed through examination of the properties of the MB scattering event shown in, for example, FIG. 1a. The data, retrieved via the transmitter and receiver configurations 10, is processed according to the IRAMS via a processor (not shown). Processors are well known in the art and will not be discussed further herein. Let (x_{i}, y_{i}) denote the twodimensional (2D) Cartesian coordinates of the spatial position of SP 2 with which the incident waveform initially interacts. Define (x_{ƒ}, y_{ƒ}) to be the spatial position of SP 3 with which the waveform that was initially reflected off of SP 2 interacts before being reflected into the receiver. Let the variable {tilde over (γ)} denotes the effective path length that the waveform traverses between the initial (x_{i}, y_{i}) and final (x_{ƒ}, y_{ƒ}) scattering centers. This path length {tilde over (γ)} is not required to be equal to the distance between the initial and final scattering centers, i.e., {square root over ((x_{ƒ}−x_{i})^{2}+(y_{ƒ}−y_{i} _{)} ^{2})}, as occurs for scattering events involving only two scattering centers. For example, triplepoint scattering events are possible, as shown in FIG. 1b. FIG. 1b shows a waveform scattering off of the initial point (x_{i}, y_{i}) of SP 4 to an intermediate location (x_{5}, y_{5}) at SP 5, before reflecting off of the final scattering center (x_{ƒ}, y_{ƒ}) at SP 6 and subsequently being collected into the receiver.
Under the definitions above, all singlereflection scattering events are characterized by {tilde over (γ)}=0. Doublereflection scattering events have {tilde over (γ)}={square root over ((x_{ƒ}−x_{i})^{2}+(y_{ƒ}−y_{i})^{2})}, whereas Npoint scattering events with N>2 are characterized by {tilde over (γ)}>{square root over ((x_{ƒ}−x_{i})^{2}+(y_{ƒ}−y_{i})^{2})}. Again, the goal of this analysis is to remove the effects of all MB scattering events, which are described by {tilde over (γ)}>0, in order to form an improved spotlight SAR image.
FIGS. 1a and 1 b reveal that the measured rangeto target of a scattering event relative to that of the coordinate origin (x=0, y=0) depends upon the observation angle θ via
The division by 2 arises since the rangetotarget is equal to onehalf of the roundtrip distance that the waveform traverses from the transmitter to the object and back into the colocated receiver. The rangetotarget spatial coordinate s is related to the waveform timeofflight t via
with c equal to the waveform speed. Given the MB rangetotarget measurements s(θ) collected over some extent of angles θ for an isolated scattering event, it is clear from Eq.(1) that it is possible to determine only the mean position of the initial and final scattering centers, i.e., {(x_{i}+x_{ƒ})/2, (y_{i}+y_{ƒ})/2}, and the total path length {tilde over (γ)} between these two points. No other information can be unambiguously gleaned.
Since only the mean position of the initial and final scattering centers can be determined uniquely, the measured rangetotarget for a Npoint scattering event with respect to the selected coordinate origin can be expressed in the form
in terms of the parameters {x≡(x_{i}+x_{ƒ})/2, y≡{y_{i}+y_{ƒ}}/2}, γ≡{tilde over (γ)}/2} of Eq. (1). Every Npoint scattering event is characterized by the parameters (x,y,γ) and traces a sinusoid in rangeangle (s,θ) that is shifted in the sdirection by a distance of γ. In contrast, each SB scattering event is characterized by the parameters (x,y; γ=0) and traces an unshifted sinusoid in rangeangle space.
Equation (3) shows that for any generic Npoint scattering event with N>1, rangetotarget measurements can yield only the values (x,y,γ). This result applies even if there are no other scattering events present. In fact, many individual scattering events having one, two, or more bounces can have identical values of (x,y,γ). Specifically, all scattering events having the same average position for the initial and final scattering centers, i.e., x≡=(x_{i}+x_{ƒ})/2 and y≡{y_{i}+y_{ƒ}}/2}, and having the same total path length {tilde over (γ)}≡2γ between the initial and final scattering centers contribute to the same (x,y,γ)point in the MB reflectivity function (MBRF) density ƒ(x,y,γ). That is, the density ƒ(x,y,γ) at a given point in (x,y,γ)space is formed by the integral of all Npoint scattering events characterized by the same values of x, y, and γ. In general, the MBRF ƒ(x,y,γ) is permitted to be a complex function, as in the modeling of the singlebounce reflectivity function ƒ_{SB}(x,y).
Consider the two different scattering events presented in FIG. 1b. The rangetotarget of the transmitted waveform is considered with respect to the parallel wave front denoted by the thin dashed lines in this figure. The line segments have been drawn so that the following relations hold:
Thus, the 2point scattering event of SPs 2 and 3, and the 3point scattering event of SPs 4, 5, and 6 have the same values of x, y, and γ within the MBRF ƒ(x,y,γ). The MBRF is the most basic representation of the regionofinterest which can be reconstructed based upon phase history E measurements that include contributions due to MB scattering events.
The measured MB projection function g(s,θ) is determined by the MBRF ƒ(x,y,γ) via
with δ( . . . ) the standard Delta function defined by A(s)≡∫ds′A(s′)δ(s′−s). That is, Eq. (7) is merely the integral of all rangetotarget projections of MB scattering events, as determined by Eq. (3). The integral over dγ in Eq. (7) applies from −∞ to ∞, provided that the MBRFƒ(x,y,γ) is constrained to be equal to zero for γ<0. If the MBRF density contains only singlebounce effects, i.e., ƒ_{SB}≡(x,y)ƒ(x,y; γ=0), then the MB transformation equation (7) reduces to the conventional Radon transform
Define the 1D Fourier transform of the function g(s,θ) with respect to s, applied independently for each observation angle θ, i.e.,
G(ξ,δ)≡∫_{−∞} ^{∞} dsexp(−j2πsξ)g(s,θ). (9)
The notation g(s,θ)⇄G(ξ, δ) expresses this relationship. The measurements for G(ξ,θ) in Eq. (9) are collected over some finite interval with respect to the spatial frequency variable, i.e., ξ_{min}≦ξ≦ξ_{max}, and some finite interval in the observation angle, i.e. θ_{min}≦θ≦θ_{max}. The spatial frequency coordinate , is related to the waveform temporal frequency coordinate τ via
so that the waveform temporal bandwidth is
Inserting Eq. (7) into Eq. (9) and interchanging the order of integration gives
Evaluation of the delta function yields the forward MB transformation between the MBRF ƒ(x,y,γ) and the phase history measurements G(ξ,δ):
This equation implicitly assumes that the MBRF ƒ(x,y,γ) is invariant over the extent of measurement angles a and spatial frequencies 4 under consideration. In a similar fashion, the omission of MB effects yields the corresponding SB result:
with G_{SB}(ξ,θ)⇄g_{SB}(s,θ).
There are two different classes of techniques for inverting Eq. (14) in order to obtain the SB reflectivity function density ƒ_{SB}(x,y) given the measured projections G_{SB}(ξ,θ). The first class of methods invokes the projectionslice theorem. However, it is not clear that these projectionslice techniques can be utilized to invert the particular MB transformation equation (13) of the present embodiment. In particular, the measurements G(ξ,θ) are collected by varying the values of only two variables, i.e., ξ and θ. In contrast, the threedimensional (3D) Fourier transform of the desired MBRF density ƒ(x,y,γ) requires an appropriate set of projection measurements over three dimensions, whereas there exist only two in the current example. Therefore, the application of projectionslice methods is not applied to invert the MB transformation equation (13) in the present embodiment. One skilled in the art recognizes that the application of the projectionslice method may be used as an alternative to inversion in an alternative embodiment, wherein the data consists of threedimensional projection measurements.
The second class of inversion methods used to reconstruct the SB reflectivity function ƒ_{SB}(x,y) utilizes backprojection functions. Back projection functions reconstruct ƒ_{SB}(x,y) based upon the measured functiondomain projection function g_{SB}(s,θ). Alternatively, a frequencydomain backprojection technique may be used to reconstruct ƒ_{SB}(x,y). The present embodiment applies backprojection techniques in order to reconstruct the MBRF ƒ(x,y,γ) based upon the phase history measurements G(ξ,θ).
Alternatively, for a continuousdomain solution, the inversion of the MB projection equation begins with the following definition of the backprojection operation applied to the MB measured transformdomain projections G(ξ,θ):
The limits of integration in Eq. (15) is chosen to be consistent with the collected transformdomain measurements, i.e., ξ_{min}≦ξ≦ξ_{max }and θ_{min}≦θ≦θ_{max}, as described above.
Substituting Eq. (13) for G(ξ,θ) into the back projection function b(x,y,γ) of Eq. (15) and interchanging the order of integration gives
in terms of the kernel function λ(x,y,γ) is defined by
Thus, the MB backprojection finction b(x,y,γ) can be expressed in terms of the MBRF ƒ(x,y,γ) via
In this equation, the symbol * denotes the 3D convolution operation. The 3D Fourier transform operator F_{3 }applied to both sides of Eq. (18) gives the following inversion technique for obtaining the MBRF ƒ(x,y,γ), given the transformdomain measurements G(ξ,θ):
Here, the operator F_{3} ^{−1 }denotes the inverse Fourier transform in three dimensions. In Eq. (19), the backprojection function b(x,y,γ) is computed by inserting the transformdomain measurement function G(ξ,θ) into Eq. (15). A solution for ƒ(x,y,γ) exists provided that the 3D Fourier transform of the kernel λ(x,y,γ) has no zeros in the regionofinterest. This constraint involved in obtaining the MBRF solution is more transparent in the discretedomain, as shown in the following subsection.
SAR image reconstruction requires that the MBRF ƒ(x,y,γ) be determined on a discrete grid in (x,y,γ) space, i.e., {x_{l}}_{l=1, . . . ,L}, {y_{m}}_{m=1, . . . ,M}, and {y_{n}}_{n=1, . . . ,N}. This grid is not required to have equal intervals between grid points in each of the three dimensions of (x,y,γ)space. However, the assumption of a uniform grid will be shown to reduce the required computation time, even if each of the three dimensions has a different resolution. This analysis generates an estimate of the MBRF matrix f(l,m,n), which is the discrete approximation of the MBRF density ƒ(x,y,γ).
The measurements arising from the coherent processing of the received radar waveform are transduced at a finite set of discrete spatial frequencies {ξ_{p}}_{p=1, . . . ,p }and a finite set of discrete observation angles {θ_{q}}_{q=1, . . . ,Q }relative to the defined coordinate system. This grid in (ξ_{p}, θ_{q})space is not required to be uniform, just as the (x_{l},y_{m},γ_{n})space of the MBRF matrix f(l,m,n) can be nonuniform. Thus, the measured phase history data are determined by the discrete domain measurement vector G(p,q) as an approximation of the density G(ξ,θ).
Given the definitions for the vectors f(l,m,n) and G(p,q), the MB transformation equation (13) has the discrete form
in terms of the matrix of phase functions
Equation (20) can be expressed in the basic form of the algebraic reconstruction problem
where F is the vector of unknowns of length U≡L×M×N, the variable G is a vector of measurements of length V≡P×Q and if is the U×V matrix of the vectors giving the projection of an (l,m,n) object space point onto (p,q) measurement space. The requirement that V≧U must be satisfied in order to obtain a solution for F, otherwise, Eq. (22) is undetermined.
A method for obtaining a solution of Eq. (22) is via the pseudoinverse
with the superscript H denoting the Hermitian transpose. The pseudoinverse gives the unique leastsquares solution provided that the square matrix H^{H}H is nonsingular. For cases in which the matrix H^{H}H is illconditioned, either diagonal loading or singularvalue decomposition can be applied. In addition, related techniques, known to those skilled in the art, can be applied for cases of additive noise.
The matrix H^{H}H is of size U×U, so that the computation of the matrix inverse in Eq. (23) is of order O(U^{3}), which is the ratedetermining step. However, techniques of order O(U×V) can be applied if the desired MBRF vector f(l,m,n) is computed on a uniform grid in (x_{l},y_{m},γ_{n})space. To develop these techniques, the backprojection vector is defined via
Inserting Eq. (20) for G(p,q) into Eq. (24) and interchanging the order of the summation gives
in terms of the matrix kernel
with
For the case of a uniform grid in (x_{l},y_{m},γ_{n})space, i.e., {x_{l}=lδ_{x}}_{l=1, . . . ,L}, {y_{m}=mδ_{y}}_{m=1, . . . ,M}, and {γ_{n}=nδ_{γ}}_{n=1, . . . ,N}, with δ_{x}, δ_{y}, δ_{γ}, all constants, Eq. (27) reduces to
Thus, Eq. (25) has the form
with
and the phase matrix
Here, the symbol “*” denotes the 3D discrete convolution. The convolutional structure of Eq. (29) permits the inversion of f(l,m,n) to be computed in FFTtime of order O(U log_{2}(U)). Thus the ratedetermining step is the computation of the backprojection vector b(l,m,n) given the measurement phase history vector G(p,q) via Eq. (24), which is of order O(U×V). Alternatively, functiondomain backprojection operations can be performed in order O(U log_{2}(U)), such that in an alternative embodiment of the present invention, the transformdomain backprojections applied to MB image reconstruction described herein, can also can be implemented in order O(U log_{2}(U)).
The SB analog of Eq. (29) has the form
with
and
Ω_{SB}(l,m,n,p,q)≡[lδ _{x }cos(θ_{q})+mδ _{y }sin(θ_{q})]ξ_{p}. (34)
Here, the symbol “*” denotes twodimensional (2D) discrete convolution. In Eq. (32), b_{SB}(l,m) is determined by
in terms of the matrix of SB phase functions
Since the measurements G(p,q) are determined by the MB transformation Eq. (20), the SB reconstruction technique of computing f(l,m) based upon Eqs. (32)(36) will contain undesirable artifacts unless contributions due to MB scattering events are absent in the measurement phase history data G(p,q).
The previous paragraphs have developed a technique for MB image reconstruction by deriving the 3D convolution equation (29), which is equivalent to the matrix form of Eq. (25). The existence of a wellbehaved solution for the MBRF f(l,m,n) requires that the 3D HermitianToeplitz matrix {overscore (Λ)}(l,l′,m,m′,n,n′)=Λ(l−l′,m−m′,n−n′) be nonsingular. The constraint is satisfied if the vectors h_{l,m,n}, (p,q)≡Ω(l,m,n,p,q) are all linearly independent functions of the phase history measurement indices p and q. This constraint is satisfied if one or more of the elements of the normalized phase vectors a_{l,m,n}<(p,q)≡Ω(l,m,n,p,q) of Eq. (31) change by unity or greater between adjacent cells in imagespace (l,m,n). The normalized phase vectors a_{l,m,n},(p,q) satisfy this requirement, and thus yield a wellbehaved solution to Eq. (29) or (25), if the following conditions apply.
First, consider the fact that spotlight SAR phase history data is typically collected over a narrow extent of measurement angles Δ_{θ}≡θ_{max}−θ_{min}<<1, here considered about θ=0 without loss in generality. For a powerseries expansion about θ=0 through only first order in θ, the factor cos(θ_{q}) in the normalized phase of Eq. (31) is approximately unity. Thus, the dependence of the alongrange coordinate x in Eq. (31) is identical to that of the MB coordinate γ, so that the effects of x and γ cannot be differentiated through first order. Therefore, a wellbehaved MB reconstruction requires that the extent of measurement angles Δ_{θ} be sufficiently large that second order effects become significant. The required second order expansion of Eq. (31) has the form:
In order to give a desired imagespace resolution (δ_{x},δ_{y},δ_{γ}), the extent of measurements in spatial frequency Δ_{ξ} and in angle Δ_{θ} must be selected to be large enough so that the zeroth, first, and second order terms in Eq. (37) all change by unity or greater between adjacent imagespace cells (l,m,n). The change in the zeroth order phase term {lδ_{x}+nδ_{y}}ξ_{p }of Eq. (37) exceeds unity between adjacent cells in both the x and γ dimensions over the extent of spatial frequencies if
which reduces to the wellknown SB relation Δ_{ε}˜δ_{x} ^{−1}.
The change in the first order phase term mδ_{y}θ_{q}ξ_{p }of Eq. (37) exceeds unity between adjacent cells in the y dimension over the extent of measurement angles if
with ξ_{cƒ} equal to the center spatial frequency. This relation is identical to that of the SB case.
The change in the second order phase term—(½)lδ_{x}θ_{q} ^{2}ξ_{p }of Eq. (37) exceeds unity between adjacent cells in the x dimension over the extent of measurement angles if
Thus, the first and second order constraints of Eqs. (39) and (40) are combined to yield
Δ_{θ}≧max{{ξ_{cƒ}δ_{y}}^{−1},{square root over (2{ξ_{cƒ}δ_{x}}^{−1})}}. (41)
For the case of identical imagespace resolutions in each dimension, i.e., δ≡δ_{x}=δ_{y}=δ_{γ}, the required measurement extents reduce to
Therefore, the extent of measurement frequencies required for the MB case is unmodified from that of the SB case. However, the extent of measurement angles is greater for the MB case than for the SB case.
In order to derive an expression for the measurement intervals δ_{ξ} and δ_{θ} required for a desired imagespace extent (Δ_{x}≡Lδ_{x},Δ_{y}≡Mδ_{y},Δ_{γ}≡Nδ_{γ}, consider the normalized phase of Eq. (37) with θ_{q}=_{q}δ_{θ} and ξ_{p}=ξ_{cƒ}+pδ_{ξ}, i.e.,
The change in the normalized phase of Eq. (44) must be unity or less between adjacent measurement samples δ_{ξ} and δ_{θ} in order to obtain a wellbehaved solution. The change in the zeroth order term of Eq. (44) is unity or smaller over the imagespace extents in the x and γ dimensions if
The first order term of Eq. (44) is unity or smaller over the imagespace extents in the y dimension if
whereas the second order term does so for
The combination Eqs. (46) and (47) gives
For equal extents in the x and y spatial dimensions, i.e., Δ≡Δ_{x}=Δ_{y}, Eq. (48) reduces to
which is identical to that of the SB case.
The SAR image reconstruction processes developed above are applied to simulated MB phase history data with the following results. FIG. 2a presents an example containing a number of assumed scattering center reflectivities. The colocated transmitter and receiver (not shown) are located at the top of each of the figures, transmitting and receiving down and up the paper in a lengthwise direction. This plot corresponds to the γ=0 plane of the MBRF ƒ(x,y,γ). In addition, it is assumed that there exists a MB coupling within the four sets of closelyspaced scattering centers located in the four corners 12, 14, 16, and 18 of FIG. 2a. For example, there is a contribution to the MBRF ƒ(x,y,γ) corresponding to the two scattering centers at (x,y)=(12, 3) and (12, 5) located in corner 12 of the truth plot in FIG. 2a. Specifically, the γ=1 plane of the MBRF shown in FIG. 2b contains a contribution at the mean position (x,y)=(12, 4) between these two scattering centers, in accord with the distance 2γ=2 between these two scattering centers. Likewise, the truth scattering centers at (x,y)=(12, 13) and (14, 13) in the lower right corner 14 of FIG. 2a yield the γ=1 MB contribution at (13, 13) in FIG. 2b. Similarly, the truth scattering centers at (x,y)=(3, 5) and (5, 3) in the upper left corner 16 of FIG. 2a have a MB coupling length of 2γ=2{square root over (2)}, thus giving linearlyinterpolated contributions at (4, 4) in the γ=1 and γ=2 planes of FIG. 2b and 2 c.
The three coupled scattering centers at (x,y)=(3, 12), (5, 12) and (5, 14) in the upper right corner 18 of FIG. 2a give three different doublebounce contributions corresponding to (x,y,γ)=(4, 12, 1), (5, 13, 1), and (4, 13, {square root over (2)}), similar to the doublebounce couplings in the other three corners 12, 14, and 16 of FIG. 2a. In addition, the upper right corner 18 scattering centers yield various triplebounce events. For example, the scattering sequence (5, 14)→(5, 12)→(3, 12) gives the MB contribution at (x,y,γ)=(4, 13, 2) in FIG. 2c, since the total path length between the initial point (5, 14) and the final point (3, 12) is 2γ=4. Likewise, the scattering sequence (5, 14)→(3, 12)→(5, 12) gives a contribution for (x,y,γ)=(5, 13, {square root over (2)}+1), and the sequence (3, 12)→(5, 14)→(5, 12) yields (x,y,γ)=(4, 12, {square root over (2)}+1).
The MBRF ƒ(x,y,γ) for γ≠0 in FIGS. 2b2 d is based upon the true path lengths between the scattering centers of FIG. 2a. However, the coupling strengths between the scattering centers were selected in an ad hoc manner to be equal to that of the truth scattering centers themselves. Nevertheless, the MB techniques described herein apply generically for any specific quantitative rules governing the MB coupling strengths.
The true values of the MBRF ƒ(x,y,γ) shown in FIGS. 2a2 d are applied within Eq. (20) in order to simulate the measurement function G(ξ,θ) shown in FIGS. 3a and 3 b. The measurement functions shown in FIGS. 3a and 3 b are coherent measurements, resulting from a quadrature demodulation of the received waveform, i.e., the reflected wave form. The xaxis is the measure of angle in radians, while the y, axis is spatial frequency. The measurement functions of FIGS. 3a and 3 b, when inverted using the single bounce theory, give the result shown in FIG. 4. These same measurement functions, when inverted using the process of the present invention, give the results discussed below with reference to FIGS. 5a5 d, creating not only an identical replica of the desired image but, also creating higher delay planes or multiple bounce image planes that contain the information rich echo portions of the reflected wave form. The signaltonoise ratio in this embodiment is approximately 40 dB. As stated above, noise and interference play no greater a role in the IRAMS processing than in the single bounce processing. Consequently, one skilled in the art recognizes the standard signaltonoise limitations that are applicable to IRAMS processing.
The results of Eq. (37)(49) are used to determine the required sampling intervals and extents of both the spatial frequency ξ and the angular coordinate θ which are necessary to reconstruct the truth MBRF ƒ(x,y,γ) of FIGS. 2a2 d. The desired imagespace resolution is δ≡δ_{x}=δ_{y}=δ_{γ}=1, so that Eq. (42) implies that a spatial frequency bandwidth of Δ_{ξ}=1/δ is sufficient for a wellbehaved MB reconstruction. SAR measurements are collected over a narrow extent of measurement angles, which is chosen to be Δ_{θ}˜0.2 rad˜11.5° in this particular example. Then the spatial center frequency is selected to be ξ_{cƒ}=2/Δ2/θ≅45, consistent with δ=1 and Δ_{θ}0.2 rad in Eq. (43). The desired reconstruction extents are given by Δ_{x}=Δ_{y}=16 and Δ_{γ}=4 for this example. Thus, Eq. (45) implies that the required sampling interval in spatial frequency is δ_{ξ}=1/20, and Eq. (49) gives a required angular sampling interval of δ_{θ}≅1/720 rad≅0.08°.
FIGS. 5a5 d show the application of the MB inversion of Eqs. (24) and (29) in order to estimate the MBRF ƒ(x,y,γ) based upon the simulated phase history data G(ξ,θ) of FIGS. 3a and 3 b. A shown, the γ=0, γ=1, γ=2, and γ=3 planes of FIGS. 5a5 d give an approximately identical reconstruction of the desired scattering center locations and strengths of FIG. 2a. Further, the extracted MB scattering events located within the γ≠0 planes can be used to augment the feature vectors applied to, for example, target recognition analysis.
For the sake of comparison, the desirable MB image reconstruction of FIGS. 5a5 d is compared with that based upon the assumption of only SB scattering events. FIG. 4 presents the result of the SB inversion technique of Eqs. (32) and (35) applied to the simulated MB phase history data of FIGS. 3a and 3 b. Clearly, the SB technique reveals a significant degradation in the estimation of the object scattering centers of FIG. 2a. Although the object locations and strengths of the various scattering centers of FIG. 2a are wellreproduced, FIG. 4 contains many additional spurious scattering centers that arise from the MB contributions to the simulated phase history data of FIGS. 3a and 3 b. Whereas the MB inversion technique accurately reconstructs a given MB scattering event characterized by (x,y,γ), the SB inversion method incorrectly projects such a point to (x+γ,y) for phase history data measured over a narrow angular extent about θ=0. For example, the MB scattering event of (x,y,γ)=(12, 4, 1) of FIG. 2b is incorrectly projected to the point (x,y)=(13, 4) within the SB estimate of FIG. 4 and similar incorrect results apply to the other MB scattering events.
The ability to form a wellbehaved MB inversion depends entirely upon the coordinate grids {x_{l},y_{m},γ_{n}} and {ξ_{p},θ_{q}} selected for the desired MBRF f(l,m,n) and the measurements G(p,q). That is, the specific number of scattering centers and their MB coupling strengths within the true MBRF are irrelevant to the inversion process, so that examples containing a much higher density of scattering events can also be generated. The particular example presented herein was selected in order to demonstrate the most salient effects in the MB and SB image reconstructions.
The embodiment described above may be extended to include data from a higher number of spatial dimensions. Define r to be a vector in an Ndimensional vector space. Likewise, define k to be a unit vector in this ND vector space. The N=2 case presented above characterizes the spatial vector via r≡[x,y]^{T}, with the superscript denoting vector transpose. Similarly, the unit vector for N=2 is expressed in terms of the polar coordinate angle θ, i.e., k≡[cos(θ), sin(θ)]^{T}. Likewise, the N=3 case can be developed by defining the spatial vector r≡[x,y,z]^{T }and the corresponding unit vector k≡[cos(θ) sin(ø), sin(θ) sin(ø), cos(ø)]_{T }in terms of the spherical coordinate angles ø and θ.
The generic ND form of the MB projection equations (7) is
The corresponding ND form of the MB transformdomain projection equation (13) becomes
In a similar fashion, the ND form of the MB backprojection equation (15) is
This ND form of the backprojection function b(r,γ) can be expressed in the following convolutional form
in terms of the kernel function
Thus, the inversion for the MBRF yields
A similar set of generic ND equations can be developed in the discrete domain.
The multiplebounce inversion technique yields excellent reconstructions of the multiplebounce reflectivity function ƒ(x,y,γ). In fact, these multiplebounce inversion methods will not degrade the reconstructed image for cases in which the phase history measurements G(ξ,θ) contain only singlebounce contributions. In contrast, the singlebounce inversion methods of conventional spotlight SAR theory yield spurious ghosting artifacts in the image reconstruction ƒ_{SB}(x,y) for cases in which multiplebounce events contribute to the measured phase history data G(ξ,θ).
In an alternate embodiment of the present invention, FIGS. 6a6 d refer to truth reflectivity functions identical to those shown in FIGS. 3a3 d. FIGS. 7a7 b, similar to FIGS. 4a4 b, show the measurement data for G(ξ,θ) corresponding to the given ƒ(x,y,γ). The measurement data shown in FIGS. 7a7 b differs from that shown in FIGS. 4a4 b, in that the data is collected at low spatial frequencies from about ξ=0.5 to ξ=1.5 as opposed to the much higher spatial frequencies in FIGS. 4a4 b which is from ξ=45.1 to ξ=46.1. Inversion of the measurement data using the MB reconstruction process described herein results in a virtually identical replica depicted in FIGS. 9a9 d of the truth reflectivity function shown in FIGS. 6a6 d. Whereas, the inversion of the measurement data using the SB reconstruction process shown in FIG. 8 contains significant degradation due to the appearance of ghost scattering centers. The signaltonoise ratio in this embodiment is approximately 40 dB. As stated above, noise and interference play no greater a role in the IRAMS processing than in the single bounce processing. Consequently, one skilled in the art recognizes the standard signaltonoise limitations that are applicable to IRAMS processing.
The IRAMS process described herein is also applicable to nearfield collection data, wherein the wavefronts are curved. For the 2D problem above, assume that the radius and angle (in polar coordinates) of the sensor are given by R(η) and φ(η), respectively, in terms of a scalar parameter η which can vary in any fashion with regard to the attributes of the scene of interest. Then the measured rangetotarget with respect to the coordinate origin is
This equation replaces Eq. (1) for the nearfield problem. The remainder of the analysis is straightforward, wherein the integrations/summations over θ are replaced by the corresponding integrations/summations over the scalar parameter η. It can also be verified by a Taylor series expansion that the nearfield equation (56) reduces to the farfield equation (1) as the radius R(η) becomes much larger than the size of the scene of interest. Equation (56) can be further extended to apply for a nonplanar collection surface by defining the sensor location in terms of the spherical coordinates (R(η),φ(η),ω)(η)) parameterized by the scalar parameter η via
This type of modeling also permits targets to have different heights z.
Further, the location of the transmitter and receiver can be different. The required modifications for such a bistatic collection are wellknown to those skilled in the art. There are two primary modifications required. First, the measurement angle θ for a twodimensional monostatic collection (wherein the transmitter and receiver are colocated) in the farfield is replaced by the bisector angle between the angle of the transmitter and that of the receiver. Second, the measured spatial frequencies for this problem are scaled by the factor cos(β/2), with β equal to the angle between that of the transmitter and that of the receiver. Further extension of the bistatic problem to the nearfield case and to a nonplanar collection surface, as presented above for the monostatic case, can be accomplished by one who is skilled in the art.
The specific examples described above are not intended to be limiting. Alteration of the processes described above is contemplated by one skilled in the art for application of the IRAMS process to various types of reflection tomographic data, including radar, sonar, and the like.
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